5-demicube


5-demicube
Demipenteract
(5-demicube)
Demipenteract graph ortho.svg
Petrie polygon projection
Type Uniform 5-polytope
Family (Dn) 5-demicube
Families (En) k21 polytope
1k2 polytope
Coxeter symbol 121
Schläfli symbol {31,2,1}
h{4,3,3,3}
s{2,2,2,2}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png
4-faces 26 10 {31,1,1}Cross graph 4.svg
16 {3,3,3}4-simplex t0.svg
Cells 120 40 {31,0,1}3-simplex t0.svg
80 {3,3}3-simplex t0.svg
Faces 160 {3}2-simplex t0.svg
Edges 80
Vertices 16
Vertex figure 5-demicube verf.png
rectified 5-cell
Petrie polygon Octagon
Symmetry group D5, [34,1,1] = [1+,4,33]
[24]+
Properties convex

In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices deleted.

It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular hypercell), he called it a 5-ic semi-regular.

Coxeter named this polytope as 121 from its Coxeter-Dynkin diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches. It exists in the k21 polytope family as 121 with the Gosset polytopes: 221, 321, and 421.

Contents

Cartesian coordinates

Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:

(±1,±1,±1,±1,±1)

with an odd number of plus signs.

Projected images

Demipenteract wf.png
Perspective projection.

Images

orthographic projections
Coxeter plane B5
Graph 5-demicube t0 B5.svg
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph 5-demicube t0 D5.svg 5-demicube t0 D4.svg
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph 5-demicube t0 D3.svg 5-demicube t0 A3.svg
Dihedral symmetry [4] [4]

Related polytopes

It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform polytera (uniform 5-polytope) that can be constructed from the D5 symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.

5-demicube t0 D5.svg
t0(121)
5-demicube t01 D5.svg
t0,1(121)
5-demicube t02 D5.svg
t0,2(121)
5-demicube t03 D5.svg
t0,3(121)
5-demicube t012 D5.svg
t0,1,2(121)
5-demicube t013 D5.svg
t0,1,3(121)
5-demicube t023 D5.svg
t0,2,3(121)
5-demicube t0123 D5.svg
t0,1,2,3(121)

References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • H.S.M. Coxeter:
    • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Richard Klitzing, 5D uniform polytopes (polytera), x3o3o *b3o3o - hin

External links


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